I'm looking for a textbook to help me with understanding the geometry of Galilean relativity and the Galilean group. The reason is that I tried going through V.I. Arnold's Mathematical Methods, but got stuck on the very first chapter because I'm just not familiar enough with affine geometry or group theory. So I want a good textbook on affine geometry, preferably using a coordinate-free approach, which goes over some physical applications including the Galilean group. Any suggestions?